Perhaps a doc bug. But an empty attribute can also be set as e.g. controls="". So

use HTML::Tiny;
my $co = HTML::Tiny->new;
my $filename = 'theme1';
print $co->tag ('audio', { controls=>"", type => "audio\/mpeg", src =>
+ "$filename.mp3"});
print "\n";
[download]

produces

<audio controls="" src="theme1.mp3" type="audio/mpeg"></audio>
[download]

which ought to have the same effect.